Statistical Concepts Everyone Should Know

class: center, middle, inverse, title-slide

.title[
# Statistical Concepts Everyone Should Know
]
.subtitle[
## Statistics for Life
]
.author[
### John Slough 
for The John & Calvin Podcast
]

---

## 1. Mean vs Median

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---

## 1. Mean vs Median

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---

## 1. Mean vs Median

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---

## 1. Mean vs Median

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---
## Mean vs Median as Skew (σ) Rises

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---

## 1. Applications of Mean vs Median

- **Wealth and Income**  
  *"Average net worth increased"* — but mostly billionaires; median person saw little change.

- **Real Estate Prices**  
  *"Average home price is \$800,000"* — driven by a few luxury sales; median much lower.

- **Medical Costs**  
  *"Average hospital bill is \$25,000"* — rare extreme cases inflate the mean.

- **Life Expectancy**  
  *"Average lifespan is 40 years"* — high infant mortality lowers the mean dramatically.

- **Response Times for Emergency Services**  
  *"Average ambulance arrival time is 8 minutes"* — median is often much shorter.

- **GDP per Capita**  
  *"GDP per person is \$60,000"* — but the median citizen earns far less in many countries.

- **Asteroid Impact Energy**  
  *Mean impact = “city-killer”* — dominated by once-in-a-million-years giants; median impact is a harmless fireball.

- **Software Bug Fix Times**  
  *"Average bug fix time is 15 days"* — most fixes happen quickly; a few linger for months.

---

## Mean or Median? &mdash; It Depends on the Shape of Your Data

- **Symmetric (bell-shaped) distributions** 
 *Mean ≈ Median* → either works 
 (heights, random errors, test scores in large classes)

- **Skewed distributions (long tail on one side)** 
 **Use the Median** &nbsp;– it ignores extreme outliers 
 (wealth, medical bills, wait times)

- **Heavy-tailed or “black-swan” data** 
 Median or percentiles are safest; mean can explode 
 (earthquake energy, insurance losses)

- **Bimodal / multi-cluster data** 
 No single centre makes sense → show the full distribution or the two modes 
 (commuter travel times, exam grades with pass/fail peaks)

.small[
*Rule of thumb ▶ If a handful of extreme observations could swing the result, report the **median** (and a spread measure) instead of the mean.*
]

---

### From “What’s the Average?” → “What’s the Distribution?”

- **Why the mean mis-led us:** the income data are **skewed**, so a few extreme values dragged the mean away from the typical person (≈ median).

- **What really matters is the full shape** of the data—its **distribution**.

- **Normal distribution (bell curve)**  
  - Symmetric, mean = median = mode.  
  - Central Limit Theorem ⇒ sample means often look normal.

- **But many real-world data aren’t normal:**  
  - Incomes/wealth → **log-normal / Pareto** (long right tail)  
  - Failure times → **Weibull / Exponential** (many early, few late)  
  - Web traffic, social-media posts, bug lifetimes … often heavy-tailed

.small[
**Takeaway ▶** Before quoting a single “average,” look at the distribution.  
If it’s skewed or heavy-tailed, report the **median or percentiles** and pick the distribution that actually fits the data.
]

---

## Bias

**Bias** is the difference between the expected value of an estimator and the true value of the parameter it estimates.

Formally:

$$
\text{Bias}(\hat{\theta}) = \mathbb{E}[\hat{\theta}] - \theta
$$

Where:
- `$\hat{\theta}$` = the **estimator** (your calculated estimate)
- `$\theta$` = the **true parameter** (the real value you want)
- `$\mathbb{E}[\hat{\theta}]$` = the **expected value** of the estimator (its long-run average over many samples)

- If Bias = 0 → the estimator is **unbiased**.

- If Bias ≠ 0 → the estimator is **biased** (systematically too high or too low).

---

Bias doesn’t just shift your first guess.
It filters what you see next — making you even more biased over time

Selection bias: The sample is not representative of the population.

Omitted variable bias: Leaving out a variable that influences both the dependent and independent variables.

Measurement bias (or information bias): Errors in how data is collected or recorded.

Survivorship bias: Only analyzing "survivors" or those who remain, ignoring those who dropped out or failed.

Recall bias: Errors because people remember things inaccurately (common in surveys and retrospective studies).

Observer bias: Researcher's expectations subtly influence measurements or observations.

Publication bias: Studies with "positive" results are more likely to be published than "null" or "negative" results.

---

## Bias

### The Self-Reinforcing Feedback Loop of Bias

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---

### Consequence of the Self-Reinforcing Feedback Loop of Bias

Conceptual Edition

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---

### Consequence of the Self-Reinforcing Feedback Loop of Bias

Carnivore Edition

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---

### Consequence of the Self-Reinforcing Feedback Loop of Bias

Vegan Edition

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